Infinity

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Denote the elements of the field $F_{4}$ by $\{0, 1, w, w + 1\}$, where $w$ satisfies the following equation with coefficients in $F_{2}: w^{2} + w + 1 = 0$. Infinity is a recurrent double sequence defined by $a(i, 0) = a(0, j) = 1$ and $a(i, j) = f(a(i, j-1), a(i-1, j-1), a(i-1, j))$, where $f(x, y, z) = x^{2} + (w + 1) y^{2} + z$. This recurrent double sequence can be also obtained using a system of substitutions of type 2 -> 4 with 5 rules.

What we present here are only the rules, renormed as system of substitutions of type 1 -> 2. The 2 × 2 minors appearing in the double sequences in even positions are replaced by simple squares, introducing for every of the 5 occurring 2 $\times$ 2 minor a new color. The system of substitutions of type 1 -> 2 got this way is called skeleton of the original system of substitutions. What you see here is “the skeleton of Infinity”.

Substitution Rule

Rule Infinity

Patch

Patch Infinity